Miller, 26, whose real name was Malcolm James McCormick, was found unresponsive in his home in the Studio City neighbourhood on September 8. The cause of death has not yet been revealed.
"I adored you from the day I met you when I was nineteen, and I always will. I can't believe you aren't here anymore. I really can't wrap my head around it. I'm so mad, I'm so sad. I don't know what to do," Grande wrote.
"You were my dearest friend. I'm so sorry I couldn't fix or take your pain away. I really wanted to. The kindest, sweetest soul with demons he never deserved. I hope you're okay now. Rest," she added.
Grande also shared a video which shows the couple laughing together as he asks her to stop filming him when he is telling a story.
Miller had long suffered from substance abuse and was even vocal about it. In his deeply personal 2014 mixtape "Faces", the rapper had addressed his drug addiction.
Miller went through a well-publicised break-up with Grande earlier this year.
After his death, Grande had shared a black-and-white photograph in tribute to Miller, but Saturday's post is the first time she has commented on his death.
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